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Representation Theory

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$U (\mathfrak{g})$-finite locally analytic representations


Authors: P. Schneider, J. Teitelbaum and Dipendra Prasad
Journal: Represent. Theory 5 (2001), 111-128
MSC (2000): Primary 17B15, 22D12, 22D15, 22D30, 22E50
DOI: https://doi.org/10.1090/S1088-4165-01-00109-1
Published electronically: May 18, 2001
MathSciNet review: 1835001
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Abstract:

In this paper we continue our algebraic approach to the study of locally analytic representations of a $p$-adic Lie group $G$ in vector spaces over a non-Archimedean complete field $K$. We characterize the smooth representations of Langlands theory which are contained in the new category. More generally, we completely determine the structure of the representations on which the universal enveloping algebra $U(\mathfrak g)$ of the Lie algebra $\mathfrak g$of $G$ acts through a finite dimensional quotient. They are direct sums of tensor products of smooth and rational $G$-representations. Finally we analyze the reducible members of the principal series of the group $G=SL_2(\mathbb Q_p)$ in terms of such tensor products.


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Additional Information

P. Schneider
Affiliation: Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany
Email: pschnei@math.uni-muenster.de

J. Teitelbaum
Affiliation: Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, Illinois 60607
Email: jeremy@math.uic.edu

Dipendra Prasad
Affiliation: Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad, 211019, India
Email: dprasad@mri.ernet.in

DOI: https://doi.org/10.1090/S1088-4165-01-00109-1
Received by editor(s): August 2, 2000
Received by editor(s) in revised form: September 25, 2000
Published electronically: May 18, 2001
Article copyright: © Copyright 2001 American Mathematical Society